Deployable structures have many functional applications such as quick-erect shelters, dynamic architectural elements and expandable structures for outer space.
In general, there are two types of deployable structures: those constructed of mechanical linkages, and those based on the folding principles of origami. Linkage-based structures are comprised of slender members connected by rotary pivots, while origami-based structures are comprised of planar facets that are joined along their edges by hinges. Unlike linkage-based structures, origami-based structures provide a continuous surface, which can be useful for areas such as coverings, display surfaces and enclosures.
There is growing interest in origami-based deployable structures. However, development is still at an early phase and challenges remain to finding practical solutions. Underlying these challenges are the rigorous constraints of origami design. The designer must ensure not only that the deployed structure meets performance requirements in terms of strength and shape, but that the deployment process—whether expanding, collapsing, or deforming in some other way—is reliable and well-controlled.
To date, a salient feature of deployable origami structures is that they are open surfaces, meaning that they have an unattached perimeter. Having an unattached perimeter is generally necessary in order for folding to occur. However, this edge is often structurally weak and tends to lose synchronization with the overall folding motion. To counter this tendency, the perimeter can be joined to external supports after deployment in order to stabilize the structure.
It has been generally perceived that open surfaces—as opposed to complete enclosures having no perimeter—is a necessary condition for folding to occur. In fact, there are rigorous conditions to achieve exact solutions for foldable enclosures (also termed “flexible polyhedra”). Most importantly the Bellows Conjecture* states that “the volume of a flexible polyhedron is invariant under flexing”. This means that it is mathematically impossible to create a flexible polyhedron whose volume changes during the deformation process. [*The bellows conjecture was proved for a subset of polyhedra (I. Kh. Sabitov, 1995) and for the general case (Connelly, Sabitov, Walz, 1997).]
Mathematicians have painstakingly identified a very small number of constant-volume flexible polyhedra. It should be noted that these examples have complex construction, limited movement and are entirely impractical for real-world applications.
Nonetheless, it would be beneficial to create foldable shapes that are closed surfaces-complete enclosures with a definite inside and outside. By eliminating unsupported boundary conditions, such structures should achieve improved structural integrity. Closed shapes may also demonstrate improved mechanical movement, in an analogous manner to how closed kinematic chains (loops) are better synchronized than open chains. Beyond these technical issues, closed shapes offer potential solutions for applications such as packaging, industrial bellows and inflatable structures.